Second-order Differential Equations with Asymptotically Small Dissipation and Piecewise Flat Potentials

We investigate the asymptotic properties as t → ∞ of the differential equation ẍ(t) + a(t)ẋ(t) +∇G(x(t)) = 0, t ≥ 0 where x(·) is R-valued, the map a : R+ → R+ is non increasing, and G : R → R is a potential with locally Lipschitz continuous derivative. We identify conditions on the function a(·) that guarantee or exclude the convergence of solutions of this problem to points in argminG, in the case where G is convex and argminG is an interval. The condition Z ∞ 0 e− R t 0 a(s) dt <∞ is known to be necessary for convergence of trajectories. We give a slightly stronger condition that is sufficient.